Quantum vs. Classical Communication and Computation

TL;DR

This paper introduces a simulation technique that converts quantum algorithms into quantum communication protocols, leading to new separation results between quantum and classical communication complexities and establishing limitations on quantum speed-ups for certain functions.

Contribution

The paper presents a general simulation method linking quantum algorithms to communication protocols, enabling new separation results and bounds on quantum speed-ups for specific functions.

Findings

Quadratic separation between quantum and classical communication complexity for bounded-error models.

Exponential separation for zero-error quantum communication complexity.

Limitations on quantum speed-up for PARITY and MAJORITY functions in bounded-error models.

Abstract

We present a simple and general simulation technique that transforms any black-box quantum algorithm (a la Grover's database search algorithm) to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism. This allows us to obtain new positive and negative results. The positive results are novel quantum communication protocols that are built from nontrivial quantum algorithms via this simulation. These protocols, combined with (old and new) classical lower bounds, are shown to provide the first asymptotic separation results between the quantum and classical (probabilistic) two-party communication complexity models. In particular, we obtain a quadratic separation for the bounded-error model, and an exponential separation for the zero-error model. The negative results transform known quantum communication lower bounds to computational lower bounds in the black-box model. In particular, we show that the quadratic speed-up achieved by Grover for the OR function is impossible for the PARITY function or the MAJORITY function in the bounded-error model, nor is it possible for the OR function itself in the exact case. This dichotomy naturally suggests a study of bounded-depth predicates (i.e. those in the polynomial hierarchy) between OR and MAJORITY. We present black-box algorithms that achieve near quadratic speed up for all such predicates.